Entanglement re-examined: Since Bell got it wrong, maybe Einstein had it right
Antony R. Crofts
Department of Biochemistry and Center for Biophysics and Quantitative Biology University of Illinois at Urbana-Champaign, Urbana IL 61801
Correspondence: A.R. Crofts Department of Biochemistry University of Illinois at Urbana-Champaign 417 Roger Adams Lab 600 S. Mathews Ave Urbana, IL 61801 Phone: (217) 333-2043 Email: [email protected]
1 Bell’s treatment of the entanglement question and the outcome of subsequent experi- ments to test his hypothesis are generally taken as demonstrating that at the quantum level nature is non-local (NL). From standard quantum mechanical (QM) predicates, when coincidences are counted as a function of polarizer settings on measurement at separate stations, tests using pho- ton pairs are expected to show the same amplitude (full “visibility”) sinusoidal curves independ- ent of orientation of the photon frame. This NL behavior, claimed to demonstrate non-locality, cannot be expected from objective local (OL) treatments. Experimental confirmation of this dif- ference is presented as one of the major achievements of 20th century physics because it con- strains all subsequent theoretical discussion of process at this level to a non-local perspective. In this paper, I argue that this edifice is built on sand. Einstein, Podolsky and Rosen (EPR) started the entanglement debate by framing it in terms of a model, necessarily local, in which quantum entities have intrinsic properties (“elements of reality”). When Bell joined the discussion, the OL model he proposed precluded consideration of vectorial properties, and then limited the behavior expected, and could not match the NL expectations. In Part A of the paper, I explore through discrete math simulation, a vectorial model (vOL) following Einstein’s expecta- tions. The simulation asks what would happen if photons with defined polarization were used in typical experimental tests, and shows that when the photon frame is aligned with the reference polarizer the outcome is the same as in the NL model. This indicates that the inequalities Bell used to exclude local realism were in error. In analysis of this discrepancy, it is clear that Bell, in excluding vectorial properties, chose an unrealistic OL model. Experimental tests failed to find the observed outcome not because nature is necessarily non-local, but because the model, and justifications for a limit of ≤2 based on it, were wrong. Nevertheless, one feature expected from the NL treatment, the full-visibility on rotation of frames, is not explained by the vOL model. In Part B of the paper, I ask how well-justified the NL predicates are, and how plausible are extensions proposed to overcome tensions with classical constraints. Three predicates from QM lead to the NL expectation: (i) that the uncertainty principle requires evolution to a measure- ment context in a superposition of states; (ii) that the pairs are in a dichotomy of H and V spin states that determine their behavior at discriminators; (iii) that the matrix operations describes a real process through which pairs are actualized with vectors aligned in frame with the reference polarizer. In the first QM predicate, quantum uncertainties preclude assignment of vectorial properties to quantum entities, demanding treatment through a superposition of states. However,
2 I suggest that this is a poor starting point because the uncertainty argument, originally applied to conjugate properties of a single electron, has no logical justification when applied to the two photons of a pair, interrogated through refraction and detected separately. The ontic dichotomy of spin states of the second predicate has no basis in physics, and the spin states determine be- havior at the polarizers by assignment of vectorial agency as propensities. The assignment of a vectorial role is also unjustified. In the third predicate, by representing the propensities in binary form in the matrices, the operations implement an unnatural alignment of photon and polarizer frames. This creates a Maxwellian demon and yields outcomes that are in contravention with second law and relativistic constraints. Given ambiguities in description of protocols, the simula- tion shows that results reported may have a simpler interpretation; manipulations in state prepa- ration lead to alignment. Rather than supporting non-locality, the results suggest local realistic explanations are more plausible than the conventional NL ones. It has been suggested that difficulties with the NL treatment can be ameliorated by later theoretical developments in the wavy realm. Dirac developed quantum electrodynamics (QED) by co-opting Maxwell’s electromagnetic wave treatment to the interaction of photons with elec- trons through their fields, with the potential for action at a distance. Some theoretical sophistica- tion is required in applying the wavy treatment to process at the quantum level because photons are neutral. Feynman’s path integral approach extended the approach by considering each path as separately quantized. In effect, Maxwell’s approach could be used to calculate phase delays aris- ing from differences in path length, giving interference effects modulating probabilities for pho- tons travelling different paths. However, quantized interactions necessarily involve processes at that level, and neutral entities cannot act through fields. A more natural approach is to treat pho- tonic interactions in terms of momentum transfer. Work can be applied either in generation of a photon by displacement of an electron, or in displacement of an electron by a photon on interac- tion by absorption or refraction. As in Maxwell’s treatment, both generation and interaction are constrained by the two properties that can be measured, frequency and orientation. A full treat- ment based on a quantized transfer of momentum has previously been suggested. In effect, this momentum-based scenario is that implemented in my simulation. Maxwell’s electromagnetic perspective underlies much of present theoretical discussion, but transfer of momentum makes for a much simpler treatment, and brings new insights into processes on scales from the quantum to the cosmic, which might provide a more promising path forward.
3 INTRODUCTION The entanglement debate started when Einstein, Podolsky and Rosen1 introduced the so- called “EPR-paradox” (Einstein’s most cited paper2). This crystallized earlier arguments3-7 with Bohr over the foundations of quantum mechanics (QM), and focused the debate through consideration of how correlations between two quantum objects should be represented when they are generated through a common transition. The EPR model was of a pair of entities, correlated through a common cause, which then separated to space-like distance before measurement. EPR pointed out that if the correlated properties were intrinsic to each entity (“elements of reality”), local measurement of one would allow the observer to infer a property of the other, allowing direct determination of pairwise correlations in a population. The predicates of the Copenhagen interpretation precluded such a pic- ture8,9. Measurement at this scale introduces uncertainty arising from quantized energy exchanges, accommodated by requiring an indeterminate superposition of all possible states of entangled pairs in the evolution to separate measurement stations for analysis. In the common wavefunction describing this non-local (NL) state, partners are correlated through a dichotomy of their spin states (the phase difference), but vectorial properties are indeterminate. Determinate entities with real vectors are actu- alized on operation of the Pauli matrices, which aligns the photon (or electron) frame with the dis- criminator frame. Measurement of an object at one location then appears to cause simultaneous actu- alization of a partner with complementary properties at a distant other. This would require transfer of energy and/or information faster than light, - that “spooky action at a distance” Einstein so disliked10. Some thirty years later, in “probably the most celebrated result in the whole of twentieth-cen- tury physics”11,12, Bell13-19 analyzed a population of electron pairs generated with opposite spin9. He compared the expectations from an objective local (OL) model with those from an NL model of the same population in superposition and showed them to be different. Specifically, his OL treatment generated a linear zigzag outcome curve as a function of angle difference, while the NL model gener- ated a sinusoidal curve. The difference opened the possibility of experimental test. Later treatments extended this approach to populations of photons pairs correlated in polarization but with similar ex- pectations15,20,21, and subsequent experiments, mostly with photons, demonstrating ‘violations of Bell-type inequalities’ have provided the main underpinning for the non-local picture4,15,19,21,22. The resulting world-picture, with properties of entangled quantum states shared and dispersed though space-time, has excited physicists, puzzled philosophers, inspired sci-fi lovers, and emboldened mys- tics over the last half-century23. Much of the excitement comes from the prospect of new physics, be-
4 cause, as is widely recognized5,20,24-32, current explanations imply an observer-dependent reality gov- erned by laws that permit information-sharing over space-like distance forbidden under classical con- straints33,34. In the first part of this commentary, I discuss this problem, and show that the OL model on which distinction has been based is flawed. I demonstrate an alternative model using populations of correlated vectorial photon pairs (vOL, an application of Einstein’s model), which, contrary to expec- tations of Bell’s theorem 7-9,13,20, gives outcome curves that match those of the NL treatment applied to the same initial state. I show that this behavior is natural and fully compliant with locality con- straints, examine why the Bell-type OL treatments have excluded such outcomes, and conclude that they represent poor choices of model and treatment. This conclusion strongly undermines the claim that violation of Bell-type inequalities excludes local realistic models. In the second part, I address the questions now opened as to whether the experimental results claimed to support the non-local interpretation really do, and whether the orthodox NL approach ade- quately explains the results claimed. I question the justification for the conventional NL treatment. I demonstrate that many experimental outcomes claimed to support non-locality, either do not, or are in contradiction with foundational QM premises, but that the natural properties of the vOL model can, within ambiguities in description of protocols, account for most without any contradiction. I dis- cuss extensions of the NL treatment by co-option of Maxwell’s electromagnetic theory to quantum field approaches. I suggest that these cannot be practically applied at the elemental level because photons are neutral and cannot interact through fields. Alternative treatments that represent photons as carriers of momentum can better represent the actions of photons in quantized processes at that level.
PART A. WHERE BELL WENT WRONG The locality constraints introduced by EPR came from their assignment of properties to dis- crete entities (cf.9,19); if properties are intrinsic, information pertinent to measurement can only be ac- cessed locally, and measurement on one of a pair can have no effect on the distant other. In contrast, NL treatments require evolution of entities in a superposition of all possible states, vectorial proper- ties are neither intrinsic nor discrete, and cannot be known until measurement, and, for entangled partners, measurement of one simultaneously actualizes that and the distant other in appropriate cor- relation. Bell11,33 claimed that, under such local constraints, when vectorial correlations were tested as a function of analyzer settings, OL expectations from a population could never match NL expecta- tions13,35.
5 Bell’s original treatment13 considered electrons with opposite spin9, determined on discrimi- nation using Stern-Gerlach magnets. The treatment was refined and extended to photons and polariz- ers by Clauser, Horne, Shimony and Holt15,21 (CHSH), and later adopted by Bell, leading to the Bell/CHSH (BCHSH19,20) consensus that inspired the early experiments22,36-39. An atomic beam (Ca or Hg) was excited by arc discharge22,39, or, in later work, by lasers36-38,40, and pairs of photons with correlated orientation were generated in a two-step decay process (cascade). The pairs evolved to separate measurement stations for analysis using polarizers set to differentiate between expectations. In more recent tests, lasers have been used to excite parametric down-conversion (PDC) in non-linear crystals17,18,41-43, using setups like that in Fig. 1. These latter sources have also been exploited in more complex arrangements following similar protocols, to explore applications in quantum encryption, teleportation and computation, and to test the finer points (loopholes) arising from the entanglement debate16,17,19,44-47.
1. The BCHSH consensus. Before discussing where Bell went wrong, it is necessary to understand his logic. Even those familiar with this field might benefit from a re-cap, so a summary is provided in the SI (Section 1). I also discuss, mainly in the second part, how Bell’s thinking was also determined by his earlier analy- sis of hidden variables. Here, a brief synopsis of the technical aspects will serve to frame the argu- ment. Although Bell initially considered electrons, photon pairs have provided the context for most experimental reports and can be discussed through essentially the same framework. The early treat- ments and experiments 15,19-21 considered photons with states designated by H (horizontal) or V (ver- tical) in pairs correlated in Bell-states HV/VH or HH/VV, and measured coincidence in detection at two stations, tested at four settings of the polarizers in a setup41 such as that in Fig. 1. With α or α' and β or β' as polarizer settings respectively at stations 1 and 2, the approach considers possible coin- cidences at settings α, β; α', β; α, β'; and α', β', chosen to discriminate between outcomes expected, - the OL zigzag and the NL sinusoid. The expectations from coincidences in pairwise measurements at 19 separate stations, (Eα,β, etc.) , then give outcomes summed through:
SBCHSH = Eα,β + Eα,β' + Eα',β − Eα',β' (eq. 1a), where E values reflect normalized coincidence yields at each of the four settings. At the elemental level, a value of +1 was assigned to a photon detected in the ordinary ray, and of -1 if in the extraor- dinary ray, and each of the four Eα,β, etc. terms can have a value ±1. The sum SBCHSH is then con-
6 strained to the range ±2 so that SBCHSH ≤ 2 (eq. 1b). This OL outcome was compared with the expec- tations from the NL treatment. Bell’s NL approach is discussed in greater detail later (Part B, section 5), but started from vector projections resulting from operation of the Pauli matrices on the discriminator vectors: ⃗ ⃗ (휎⃗ ∙ 훼⃗ 휎⃗ ∙ 훽)=−훼⃗ ∙ 훽 = −cosσ (eq. 2). Using photon pairs and standard QM principles, the matrix operation generates a sinusoidal curve as a function of angle difference, σ, between polarizers at different stations, with an amplitude of ±2, invariant on rotation of the photon and polarizer frames. The invariance depends on the dichotomic correlation in spin state in the population, actualization from the superposition as photons with real vectors in alignment with the polarizer frame. Then, the yields at the polarizers are given by Malus’ law, the symmetry of the matrix operations leads to cancellation of contributions from the photon vectors, and coincidences are dependent only on angle difference, σ. Fig. 2 (see legend for explana- tion) shows a unit circle representation of such projections, leading to the observable Malus’ law val- ues at particular settings of the variable polarizer.
Equations (eqs. 1a, b) are taken as the basis for Bell’s OL treatment, and values for SBCHSH then depend on estimation of values for the Eα,β, etc. terms through a conventional application of probability theory. In the seminal treatment, as adapted for photons21, Bell first asked what would be expected from local measurement on a stochastic population. In the context of photons, this gives an elemental probability 푝 (휆, 훼), derived from the vectors of the photon polarization, λ, and of the po- larizer, α. The outcome with a population, is then given by integration, 풑 (휆, 훼) = ∫ 휌(휆)푝 (휆, 훼)푑휆 (eq. 3a), where the function 휌(휆) is a normalized probability distribution for λ. With the stochastic population considered, terms in λ would cancel out, because∫ 휌(휆)푑휆 =1 (eq. 3b) (the isotropic con- dition). As a consequence, no information about λ would be available from measurement on the pop- ulation at either station. OL expectation of coincidences could be obtained by similar integration us- ing the product of elemental probabilities at the two stations. With polarizers set at α (station 1) or β (station 2):
퐸 , ( ) = ∫ 휌(휆)푝 (휆, 훼)(휆, 훽)푑휆 = ∫ 휌(휆)푝 (휆, 훼, 훽)푑휆 = 풑 , (훼, 훽) (eq. 4), with similar equations for other polarizer settings. In equating the first and the second equations on the right of eq. 4 (RHE1 and RHE2), and in deriving 풑 , (훼, 훽), Bell concluded that, as implicit in RHE2 and the i sotropic condition, eq. 3b would come into play, so that terms in λ would also cancel here. Then, 풑 , (훼, 훽), and Eα,β (OL), etc.,
7 and hence SBCHSH, would depend only on polarizer orientations. From this he concluded that photon vectors could play no part in determining the outcome, and therefore developed an OL model in which, instead of the polarization vector, the photon correlations were represented by the scalar sign of the spin. This has been likened to an “exploding penny” model, in which the explosion separates the head side from the tail side, with the two parts hurtling apart in opposite directions48. The equa- tion he suggested,
Eα,β(OL) = p1,2(α, β) = -1 + 2σ/π (eq. 5), yielded a linear dependence on σ in the range ±1, giving a zigzag curve as a function of angle differ- ence, invariant on rotation of frames, yielding an amplitude in the range ±2 for SOL, the sum of coin- cidences at the four settings expected from eq. 1a. Since curves from the two models are different, they predict different outcomes from coinci- dence measurements at canonical angle differences along the curves, giving “inequalities” expressed as limits of either SNL ≤ 2.83 from the NL sinusoid, or SOL ≤ 2 from the OL zigzag. The same limiting value can alternatively be derived generically for any OL model12,15,19-21,49 from substitution of elemental counts of ±1 in eq. 1a above; for the four terms, this gives SBCHSH as ±2 (eq. 1b). Since the mean from integration over a population cannot exceed the maximal elemental value, the limit SOL ≤ 2 for all OL models follows from this approach. These are the first and second of Bell-type inequalities. Despite spectacular technical and theoretical advances, this treatment con- tinues to provide the consensus framework for discussion of the inequalities in the entanglement de- bate. In these terms, a “violation of Bell’s inequalities” means that the result of an entanglement test significantly exceeds the ≤2 limit expected from the OL model. Bell’s third inequality came from an OL model based on a stochastic population of correlated photon pairs retaining vectorial properties and will be discussed in detail later (Part B 1).
2. The critical experimental results. Over the years following the first results22,39, an early consensus developed based on out- comes testing readily distinguished features; the difference between Bell’s zigzag and the NL sinus- oidal curves, and the invariance of the sinusoidal outcome on rotation of the polarizer frame4. With cascade sources, technical difficulties and inefficiency of detectors (at <20% efficiency, <4% of co- incidences could be detected), required normalization of coincidence counts, and correction for “ac- cidental” counts22. The ‘inequalities’ were represented in the δ-function, which scored the difference between coincidences measured at canonical angle differences, and those expected from the OL
8 limit22,36,38. In terms of the above criteria, the NL case wins hands-down, because the OL model was shown to fail; with one exception, later disavowed (cf.50), early tests22,36,37,39,40 showed the rotational invariance expected from either model, but the sinusoidal curves expected from NL, rather than the OL zigzag. After normalization and correction, amplitudes were interpreted as showing the full visi- bility expected from NL models4,20-22,28,36,48, and these results steered the debate towards a consensus supporting Bell’s theorem49. Subsequent experiments of increasing sophistication16-18,41,51-53 using PDC sources, have refined results so as to approach closer and closer to the NL expectation of 2.83, with the ‘violation of Bell-type inequalities’ scored through standard deviations from the OL expec- tation of ≤2 (cf.36,38,41,47,54). In recent work with high-efficiency detectors (~91%), outcomes meas- ured have exceed the OL expectation without any special corrections55,56. Based on such results, the quantum theorists have concluded that no explanation in local realistic terms could be tenable. For Bell13, “…the quantum mechanical expectation value cannot be represented, either accurately or arbi- trarily closely…” by any equation in the form of eq. 4. Since in all reports the results matched the NL sinusoid, they could only be accounted for in local realistic terms by invoking “hidden variables”, re- quiring “conspiracies” between source and stations to explain the vectorial outcome, a view summa- rized in “…if [a hidden variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local…” (from Bell57, as qualified by Shalm et al.56). Various scenarios (“loopholes”19), most notably detection and communication loopholes, are dis- cussed through which such conspiracies could be enabled. Experiments, some of spectacular sophis- tication (cf.55,56,58-60), have successfully eliminated such loopholes, apparently solidifying the NL case. Shimony’s conclusion19 that “the prima facie nonlocality of Quantum Mechanics will remain a permanent part of our physical world view, in spite of its apparent tension with Relativistic locality” is widely accepted (cf. 39).
3. Simulation of a vectorial OL model (vOL) shows that the BCHSH OL model is inade- quate The BCHSH algebra seems impeccable, but something is not right. Prediction of the sinusoi- dal outcome of the NL treatment depends on application of Malus’ law to pairs actualized with real vectors. In the derivation of Bell’s OL zigzag above, vectorial information about the photon source had been excluded, so Malus’ law was not involved. His model could never generate sinusoidal curves because the vectors needed at the polarizers had been replaced by scalar spin states.
9 In order to better understand what factors might be important, typical tests17,41,53 using photon pairs (Fig. 1) have been modeled through discrete math simulation. The initial intent was to address the naïve question implicit in the above paragraph: What would be the outcome if a vectorial local realistic (LR) population of correlated photon pairs was analyzed using Malus’ law compliant polar- izers? The short answer is – the sinusoidal outcome expected from NL predicates (Figs. 2, 3A).
(i) Simulation program. The core of the program is simple. For each ‘experiment’ a population of ‘photon’ pairs is generated with each partner defined by an explicit polarization vector appropriate to program settings for source, Bell-state, correlation, etc. (the Make Light subroutine). Uncertainties are implemented where appropriate (value of photon angle for each pair in a stochastic population, variable polarizer settings, allocation of photons to beams, etc.) by use of a random number generator. Orientations of the polarizers are set after the photon population has been established (“with photons in flight”), but before discrimination, measurement and plotting of coincidences and singles counts (the Measure- ment and Plot Point subroutines). The most important difference from Bell’s scalar dichotomic model is that the photons of the vOL population measured have the intrinsic vectors Einstein would have expected. As in real experiments, yields at the ‘polarization analyzers’ are derived from discrete meas- urement of elemental outcomes; a photon appears in either the ordinary or extraordinary ray of a po- 2 larization analyzer. Discrimination at the polarizers is assumed to follow Malus’ law (I/I0 = cos φ, with I/I0, the normalized transmission, θ the orientation of the polarizer, λ the polarization vector of the photon, and φ = θ - λ) At the elemental level, Malus’ law applies statistically; the yield calculated is a probability for transmission corresponding to the BCHSH term, 푝 (휆, 훼) above. The sampling of the statistical spread is implemented by incrementing counts at detectors (Q or R at station 1, S or T at station 2, Fig. 1), based on comparison of the normalized Malus’ law yield to a random number between 0 and 1; if the yield is greater, the photon goes to the ordinary, else to the extraordinary ray. This gives the single counts. Two exceptions to this rule are in simulation of Bell’s OL model, and in mimicking the QM expectations. Simulation of Bell’s OL model is by comparing the calculated yield to 0.5 to give a binary outcome (see below); simulation of the QM expectations is by implementing the propensities of each photon implicit in the outcome of the matrix operations as summarized by Shimony19 (see Part B, Section 7). For each pair, coincidences between stations are scored, with the mean plotted to give points on the coincidence curve. Scoring of coincidences can be selected from a
10 choice among four different algorithms. With the two exceptions noted, all these components are fully compliant with strict locality constraints. At any particular setting of controls, a point on the outcome curve is determined from a popu- lation of pairs by a simple count of pairwise coincidences. Useful features of the program are diag- nostic aids. The menu bar (Fig. 2, top) includes different Run options that implement a complete set of experiments generating an outcome curve. These Run options iterate though three subroutines (Make Light, Measurement, Plot Points in the Menu bar) that implement the functions suggested, to generate a point on the curve. The user can choose from several different counting algorithms by as- signing values (count increment, true/false, ±1 in CHSH protocol) appropriate to the different algo- rithms (anti-correlation count, Boolean coincidence count, CHSH count. An emulation of the Freed- man and Clause (FC) δ-count, can also be chosen (see Program Notes for implementation). Which- ever is used, the outcome is, as appropriate to the settings, essentially independent of the algorithm. Counts at the elemental level follow the complementary symmetry expected12,19, seen experi- mentally37,38,40,61, and discussed at length by Mermin61. Elemental counts for the last population tested, and vectors relevant to measurement of each pair, can be displayed pair-by-pair in the Gadgets (top right, Fig. 3A). The pairwise counts generated at a particular setting of the polarizers show ele- mental values, - either +2 or -2 for the CHSH count, 0 or 4 for the anti-correlation or Boolean counts, which accumulate when integrated in a population to give points on a curve plotted as a function of polarizer angle difference, with the Boolean count 90o out of phase with the anti-correlation count. The range shown in the Gadgets (±2 or 0-4) might lead to some confusion, because, on a sin- gle run, the fixed polarizer is set at a particular value, and the range of the count could reflect only ' two (for example α, β and α, β ) of the four E terms contributing to SBCHSH (a range ±1 or 0–2). The range shown in the Gadgets comes from scoring both detection and non-detection, which is practica- ble with perfect detectors, and facilitates taking cross-products, but counts each coincidence twice (half of the coincidences are redundant because a non-detection always mirrors a detection). How- ever, after normalization to the two photons of a pair, the range at a particular setting (±1 for the CHSH count, etc.) is the same as the conventional scoring, and this is the default for outcome curves shown in the display panels. In each of the counting algorithms, the summation over a population represents an integration in the form of eq. 4, in which RHE1 is considered as the expectation. The outcome is determined solely from the elemental counts. The program functions at the level of observables, defined by the use of Malus’ law probabilities. No algebraic sophistication is involved in the simulation, - refractive components are treated naïvely in terms of their empirical behavior, and only linear polarization is
11 considered. Elementary trigonometry is used, but in only two contexts; (i) to determine from Malus’ law what yields would be expected at the settings used, so as to implement the statistics above; and (ii) to calculate Malus’ law compliant theoretical curves. Links to the executable program (Bell_Ineq.exe), source codes (in Microsoft Visual Basic 12), and Program Notes (also accessible as Help in the program), are available in the SI.
(ii) Program outcome. The correlations observed depend on the model for the photon population, on how the ‘polarizers’ are chosen to respond (Fig. 3 A-E), and on choice of settings. When the polariz- ers are set to implement Malus’ law, photon pairs are oriented in a common frame, and share an an- gle with the fixed polarizer, coincidence counts from a vOL population follow the full-amplitude si- nusoidal curve expected from the NL treatment (Fig. 3A, LR0 curve), not the linear zigzag expected from Bell’s OL model. Compared to the coincidence count, the CHSH count gives essentially the same curve (Fig. 3C) but offset because of the choice of ±1 for elemental values, and the anti-correla- tion count is 90o out of phase. The curves show the full amplitude (‘visibility’) expected from the NL treatment. This LR0 outcome applies at all angles for the common frame (Fig. 3 A, C, E), - a classi- cal rotational invariance. When the photon pair orientation is random (isotropic in the plane of measurement), and the polarizer function is natural, the outcome is a sinusoidal curve, which shows the invariant behavior expected by NL on rotation between the photon and polarizer frames, but with amplitude half that expected from NL (Fig. 3B). Such behavior (LR2 curve) is fully compliant with locality con- straints62,63; the reduced amplitude represents the entropic penalty on measurement of a stochastic source. As discussed at greater length in Part B 1, it corresponds to expectations from Bell’s third in- equality. With the photon source randomized and the polarizers set to Bell binary mode (equivalent in effect to Bell’s scalar choice), the correlations follow the linear zigzag expected from his OL model20, with a partitioning proportional to σ (Fig. 3D, LR1 curve). This outcome is also invariant to rotation of frames. However, in contrast to the LR2 outcome, it shows full amplitude. Note that the photon population of the simulation differs from Bell’s original model in retaining its vectorial char- acter, so the outcome here depends on forcing the polarizers to behave (unnaturally) as binary rather than Malus’ law discriminators (see above for implementation). Outcomes from other combinations are unremarkable, though not without interest. The user can set parameters for modifications to the photon state on insertion of additional refractory elements into one or the other path, or both. Different coincidence counts give the result appropriate to the
12 phase relationships tested, with different Bell states showing the expected behavior. However, with oriented photon pairs, when the polarizer frame is rotated away from alignment with the photon frame (or vice versa), the amplitude of the sinusoidal curve, as expected, decreases, and is zero when the rotation is by 45o (Figs. 2, 3E). This is in contrast with NL expectations, where full-amplitude is expected at any orientation of the photon frame, including stochastic. The NL outcome can be gener- ated by implementing a “QM simulation option” to match the effective outcome suggested by Shi- mony19, - as discussed later in Part B, this is in effect, a Maxwellian demon. For settings at which Bell-correlated pairs were analyzed, the local mean yields (the ‘singles- counts’) at the four outcome rays have the same property, - a mean probability of 0.5 (within the sta- tistical limitations of the sample), independent of photon source or polarizer settings (right-panels in Figs 3A, B, D). This is what Bell expected. An Analog option allows display in the right panel of the fractional yield differences given by Malus’ law (the sinusoidal curves in Fig. 3C, right panel, explained in the legend). The theoretical curves in the left panel can be derived from the analog distributions in several ways, but the points plotted always reflect the statistical outcome from counts of pairwise coincidences at the two sta- tions. The full visibility theoretical curves expected from the NL treatment (in effect the LR0 curves) are displayed in the left panel when frames are aligned to match that expected under NL predicates. In summary, the outcome shown in Figs. 3A and 3C are the full-amplitude sinusoidal curves expected under NL predicates (cf.16,18,28,48), generally presented as impossible from any OL treatment, but which here demonstrate a violation of Bell’s inequalities from a vectorially correlated population of photon pairs, fully consistent with OL constraints.
4. What can we learn from the simulation? Local realistic models are often presented as classically constrained and are contrasted with NL models following QM predicates. However, in the argument between Einstein and Bohr both models were QM inspired, though interpreted from different perspectives. The behaviors extracted in my vOL simulation follow from empirical treatments that are essentially classical but are quantized as expected by Einstein. No model following his expectations would be antithetical to foundational QM principles. In the simulation, all interrogative interactions (refractive discrimination at polariz- ers, half-wave plates (HWPs) etc.) involve discrete photons and full commitment of their action. In effect, for entities with intrinsic properties, all exchanges are quantized and local, but all behaviors (except the QM simulation and Bell binary options) follow empirically justified laws (see Part B for discussion).
13 The vOL approach depends explicitly on the factorizability of cross-probabilities within a scalar algebra64 (eq. 4, RHE1), generally considered as characteristic of valid implementations of the locality constraints19. The LR0 outcome shows, contrary to conventional expectations, the same full amplitude sinusoidal curve as the orthodox NL treatment. Such an outcome should not come as a surprise. For any particular setting of the fixed polarizer, there must always be a vectorial LR popu- lation (readily available using PDC) that generates the same curve as NL, - that in which the photon population is oriented and aligned with the polarizer frame, - the configuration expected under NL predicates on actualization at the time of measurement. The outcomes are then the same because the vector projections are the same (see Fig. 2). As long as the photon vectors are represented, the only condition needed for prediction of such an outcome is the alignment of frames. An important conclu- sion from this is that the difference between treatments must lie in how alignment is achieved; it is implicit in the NL matrix operations but has to be explicit under vOL constraints. With this proviso, since the curves are the same, the differences in yields at canonical values (~2.83) ‘violate Bell-type inequalities’ just as does the NL value. The conclusion that no OL outcome can match the NL expec- tation value must then be invalid; in particular, the OL limit of ≤2 must be artificial. Since the sum- mation of outcomes giving the LR0 curves is equivalent to integration in the form of eq. 4 (the RHE1), Bell’s conclusion that “…the quantum mechanical expectations cannot be represented (in that form) …” is also clearly wrong. I make no claim for originality in introducing the vOL model; it is simply Einstein’s perspec- tive applied in a vectorial context. Several previous efforts (cf.62,65-72) have arrived at similar conclu- sions, most explicitly in62,67,73. The earliest of these by Angelidis65, a protégé of Popper74, was dis- missed by Garg and Leggett49, essentially in terms of the limit of ≤2 from the second (BCHSH) ine- quality (eq. 1 above). Their brief paper was selected by the editors as representative of a much wider community that responded similarly. This consensus reflected a confidence in the expectations of Bell’s theorem, reinforced by its apparent validation in contemporary experiments22,36-38,40. The rejec- tion was perhaps understandable in a historical context, reflecting wide acceptance of the Copenha- gen interpretation. Similar dismissals of all later claims have been justified by the same rationale. However, from the above, perhaps confidence in this dismissal was misplaced.
5. Bell’s three mistakes
Note that in Bell’s analysis summarized above, the value of SBCHSH depends only on the angle difference between polarizers, σ. The properties of the photons in the source population were not
14 considered because he concluded that the vectors were “hidden variables”35, and could not be ac- cessed locally. Without vectors for the photon source, discussion is constrained to Bell’s framework, - a mental box that has effectively precluded consideration of Einstein’s model. a) Local measurements. For stochastic sources, as Bell pointed out, the mean local yields would necessarily be independent of polarizer orientation. All early treatments involved such 22 sources; the outcome was measured in the R1/R0, etc., terms of Freedman and Clauser , and this was use in later reports using Ca-cascade sources, and also by Fry and Thompson36 with a 200Hg-cascade excited by laser. It is simulated in the singles-counts of 0.5 (see right panels of Figs. 2A, B, D), and is diagnostic of an isotropic condition in the plane of measurement. Bell’s first mistake was to draw the wrong conclusion from this outcome, - that vectorial properties of photons could not be involved in the behavior seen on local measurement, and should therefore be excluded consideration. b) The singles-count at each station are accounted for by natural behavior at the quantum level. Although values for individual photon vectors are lost in the mean, the behavior observed locally must access them at the elemental level. It depends on what happens at the polarizers, where the Ma- lus’ behavior law requires vectors for both photon and polarizer. The experimental outcome can then be explained naturally in terms of local vectorial properties. For elemental measurements, I/I0 (the
Malus’ law expectation, see Section 3, (i)) gives the BCHSH probability, 푝 (휆, 훼), and on integration over a population at λ, the Malus’ law yield. With a stochastic population, and a polarizer at any set- ting θ, sampling λ by integration over the hemisphere (as in eqs. 3a, b) would show a distribution of values for 푐표푠 휑 varying with λ, centered at the polarizer vector, θ, and with the mean yield of 0.5. Since the same curve is found at all values of θ, this accounts in terms of local properties for the be- havior Bell took as demanding exclusion of such properties. There are no “hidden variables” in this treatment, so their invocation in discussion is not useful, and Bell’s conclusion that vectorial proper- ties were not involved in the behavior observed was wrong. The zigzag outcome is contrary to the sinusoidal Malus’ law behavior demonstrated in 200 years of experimental work exploring polariza- tion. When using an oriented source of photon pairs with dichotomic distribution of H and V in each population (as in Bell states HV/VH or HH/VV from PDC), the mean yield in any ray is given by 0.5(cos2(θ – λ) + cos2(θ – (λ + 90o))) = 0.5(cos2θ + sin2θ) = 0.5 (eq. 3c), where θ is the orientation of the polarizer and λ that of the photon reference frame (the H photon)57,69. The partition of H and V
15 photons is isotropic because symmetrical about the reference axis, and gives in the mean, the same yield as Bell’s integration for a stochastic source. Within the above constraints the outcome is there- fore independent of source model, or specific values for θ and λ. A similar mix could be expected from either an OL or a NL state, so this outcome is of little interest in distinguishing between models. c) Expectations from comparison between stations. No vectorial correlations between stations could be predicted from the mean yields from local measurements because all information that would allow comparison of each photon to its partner is lost in the mean. Experimentally, correlations are determined from pairwise comparison of elemental measurement outcomes at separate stations. This selection is important because with pairs in stochastic orientation, correlations are conserved only on a pairwise basis. With cascade sources, the protocol was designed to select pairs by temporal coinci- dence, and choice of color filters based on properties expected from the cascade, and on direction of flight. Coincidences were maximal when polarizers were aligned, demonstrating that both photons of a pair had close to the same orientation; since these last two properties are expected from conser- vation of angular momentum in the source process, the protocol was predicated on determinate prop- erties. Correlations were either detected on-the-fly by coincidence counters, or determined from data recorded and time-tagged on-the-fly and analyzed later. Further analysis requires the pairwise data, their time of measurement, knowledge of polarizer settings, etc., but all this information is exchanged subluminally 68,69. Different approaches providing justification for the OL limit of ≤2 were outlined above: (i) Bell’s derivation from the zigzag. In deriving the zigzag13, Bell started by applying eq. 4 a stochastic source. Then, in light of RHE2, the integration through eq. 3b eliminated con- sideration of vectorial properties. These were replaced by the sign of the spin, leading to a scalar partitioning to different hemispheres (eq. 5), equivalent in effect to the binary discrimination giv- ing my LR1 curve. Peres48 suggests that “Bell’s theorem is not a property of quantum theory. It applies to any physical system with dichotomic variables, whose values are arbitrarily called 1 and -1”. However, EPR certainly considered their model to be consistent with the foundational QM principles established by Einstein over the previous three decades. If Bell’s OL treatment was “…not a property of quantum theory…”, it was because, contrary to EPR, he stripped the entities of their intrinsic vectorial properties. He thereby eliminated consideration of Einstein’s model, - Bell’s second mistake. The zigzag (as demonstrated in the LR1 curve in my simulation, Fig. 3D) would be expected for radiating scalar partners from dichotomic pairs (the “exploding penny” model), but scalar correlations could never lead to a sinusoidal curve.
16 (ii) Expectations from eq. 4 depend on the degree of order in the source. With a stochas- tic source, the outcome expected from eq. 4 is dependent only on the polarizer difference; it is independent of the setting of the reference polarizer. This behavior is a consequence of the iso- tropic condition, which justifies invocation of eq. 3b., and leads to an expectation that the out- come, 풑 , (훼, 훽), would be rotationally invariant. However, this is not a consequence of any loss of vectorial properties. This can be seen in the fact that all experimental reports have showed si- nusoidal outcome curves; these simply shows Malus’ law behavior at the polarizers, which re- quires that photons have vectors. This natural behavior is also reflected in the half-amplitude LR2 curves of the simulation, and their rotational invariance. The apparent “loss” of vectorial conse- quence is a measurement problem, a trivial epistemological issue. In this light, Bell’s third mis- take lay in extending the wrong conclusion from his first mistake (above) to interpretation the outcome expected from the count of coincidences. Both for the singles counts and the coinci- dence counts from an isotropic population (eq, 4), he interpreted the expectation that the behavior would be rotationally invariant as showing that the vectorial properties of the photons could not contribute to processes determining that behavior. His replacement of the vectorial property by the sign of the spin was an ontic surgery, in effect removing the vectorial property. This made it impossible to take into account the elemental vectorial processes involved in the behaviors. The count from integration at a single station and the integration of the pairwise differ- ences between stations involve different operations. The singles-counts come from integration at one station of elemental responses from an isotropic mix of V and H photons (eq. 1-3). Then each population measured at separate detectors gives the same singles count, no matter how the polarizer is set (Section 5 b)). In contrast, pairwise measurement involves detectors at two sepa- rate stations, and analysis in which each photon is compared (via Malus law) to its space-like separated partner. While in the singles counts, summing the elemental probabilities at a local sta- tion (eqs. 3a and 3b, or 3c) gives in the mean the same 0.5 local yields at any polarizer setting, when applied in pairwise counts, the same elemental probabilities lead to Malus’ law differences which accumulate to yield, in the mean, points on a sinusoidal curve. The Malus’ law outcome is 2 2 given by Eα,β = cos σ - sin σ = cos2σ, etc.; at the elemental level, the terms are probabilities ex- pressed in the distribution of ±1 values on detection at each station. In the mean from a popula- tion in a stochastic distribution of vectors, this generates a curve of half-amplitude, analyzed in detail in Part B.
17 With an ordered population, since eq. 3b does not come into play, although RHE1 still holds, the RHE2 term is no longer relevant, and expectation of rotational invariance from
풑 , (훼, 훽) no longer holds. With an ordered and aligned photon source, the mean from integra- tion of pairwise coincidence (in effect using RHE1) gives the LR0 curve, but the same outcome can also be derived analytically from that equation, using the four cross-products between yields calculated at the two stations (QS, RS, RT, and QT) taken for each configuration of the pair (for example VH or HV). The outcome (the green curves) depends on the degree of order in the popu- lation. For ordered populations, with the photon and polarizer frames aligned, projections from the eight comparisons above lead to the same Malus’ law outcomes as in the matrix operations of the NL approach, and differences give points following the full-visibility curve expected from this (see Fig. 2 and legend). For a stochastic population, the value for λ for each particular pair will be different. In pairwise measurements, partner is still compared to partner, but the mean will be reduced by the entropic penalty from cancellations arising from the stochastic distri- bution of values for λ to give a cos2σ curve of half-amplitude (the LR2 curves are analyzed in more detail in Part B 9). With ordered populations misaligned, the standard probability ap- proach above gives the green curves in Figs. 3E, the Malus’ law result. In all cases, the sinusoidal shape of the curve simply shows Malus’ law in action, nothing more. No natural vectorial state could generate the zigzag; no scalar state could generate the si- nusoid. (iii) Derivation of the <2 limit from the elemental count, - the BCHSH inequality. This approach, first hinted at by Clauser et al.15, further developed by Bell20,35,75 and by Clauser and Horne21, expressed with clarity by Leggett12,49 and reviewed comprehensively by Shimony19, was based on showing from the elemental coincidence values of ±1, that Eα,β + Eα,β' + Eα',β − Eα',β' from eq. 1 is limited to ±2, and noting that this constrains the mean value from a population, SOL, to ≤ 2, to set a limit for all OL models. This was then compared to the SNL = 2.83 value from the NL treatment at canonical angle differences. The maximal value of 2 defines the OL limit, the NL expectation of ≤2.83 unambiguously exceeds that limit (both features are also demonstrated in the simulation), so the conclusion that no OL model that conforms to Bell’s constraints could match the NL expectations might seem unassailable12,19. However, although the math is correct, the conclusion is wrong. It depends on two mistaken assumptions: (1) that the maximal value of
2 applies only to the OL case, and (2) that it is directly comparable to the canonical value for SNL of 2.83. That neither is true becomes obvious from examination of the outcome curves scaled to
18 four units through the S parameter (Fig. 4). The same sinusoidal curve in the range ±2cos2σ can be generated from either the vOL or NL model, (disproving (1)); and, although both the maximal value of 2 and the NL limit of 2.83 belong to the same outcome curve, they describe different properties of the curve, so are not comparable (disproving (2)). These two mistakes cannot be blamed on Bell; they have been perpetrated through their acceptance by the whole commu- nity12,29,48,49. (a) The range ±2 is a consequence of the choice of elemental values of ±1. Since cosσ varies between ±1, the same elemental values and the same maxima and minima also define the NL curve; as shown in the simulation, at appropriate alignment, the NL and vOL curves are the same. The maximal value and the properties of the curve therefore apply to both models. The S parameter can take any value in the range ±2 (from eqs. 1a, 1b):
-2 ≤ S = Eα,β + Eα,β' + Eα',β − Eα',β' ≤ 2 Although the count constrains the maximal amplitude of the outcome curve to 2, the value of a point is constrained by the ±2 limits; any difference between two points is constrained to the 4 unit range of the curve. (b) That the OL limit of ≤2 is problematic should then be obvious. The OL limit of 2.0 comes from a singular point, the maximum of the curve. In contrast, the NL limit comes from differences between two points on the curve. At any pair of canonical values for σ, those two points fall symmetrically about 0 within the scale range of ±2 (right scale pertaining to the CHSH count in Fig. 4). For example, at canonical settings of 22.5 and 67.5 (as in Fig. 4), the S values are 1.414 and -1.414, with the difference of ~2.83 applying to both models. Exclusion based on the comparison between the maximal value and the difference is then obviously absurd. (The points on Bell’s zigzag are 1.0 and -1.0, giving a limit of 2 from the first inequality, but since the model is wrong, this is now of historical interest only.) (c) For any simple count of coincidences (Fig. 4, left scale, the anti-correlation count (red points) or the Boolean coincidence count of Fig. 3A), the curve will fall naturally in the range 0 - 4, and any ordered population in which frames align will give the full-visibility sinusoidal 4cos2σ curve. Values at the canonical intercepts show the same difference ≤2.83, but this is un- remarkable when compared to the amplitude limit of 4.
Limits of <2 as derived above provide no basis for discrimination between local and non- local models, - they reflect instead either a poor choice of OL model or a poor treatment or both. When the photon and polarizer frames are aligned, there is no difference between vOL and NL
19 expectations. The failure to find Bell’s OL expectations experimentally is unremarkable since it was based on an unrealistic model. d) Fitting the sinusoidal curves. When real vectors in distinct frames for photons and polarizers are used to represent values pertinent to a vOL model, the results differ from NL expectations only when the frames are misaligned. For aligned populations, the full-visibility sinusoid can be derived directly from the Malus’ law yield differences in each ray (right panel of Fig. 3C and legend). The conventional NL treatment (cf.19) in effect actualizes photons with their vectors in that same alignment. When oriented OL populations are not aligned, the more complete probabilistic analysis giv- ing the green curves (Fig. 3E) is required. In the simulation, such curves are obtained (as a function of σ) by counting the pairwise coincidences, finding the mean by the integration of eq. 4, using
RHE1, and calculation of SvOL from the sum of eight outcomes as discussed above (Part A, 5a)). An- alytically, the green curves are derived from the same eight outcomes but calculated from the Malus’ law expectations. These treatments embody all the BCHSH realistic prescriptions for OL treatments and generate simulated or theoretical curves that fit all outcomes using oriented populations. If detec- tors in both outcomes of a polarization analyzer are measured, the same eight terms contribute to the mean count from experimental pairwise comparisons. The set is formally equivalent to those in play from the matrix operation of the NL treatment applied in the plane of measurement (Fig. 2). e) Known unknowns. No elemental measurement can lead to complete specification. Elemental events involving photons are necessarily quantized, but classical statistics will still apply (see76, and Part B). The simulation demonstrates that the sinusoidal outcome curves are not a consequence of indeterminacy, or of inseparability, superposition, or any of the algebraic paraphernalia said to be re- quired to deal with QM uncertainty. As long as the “uncertainties” are distributed normally about the mean, the counts of elemental pairwise coincidences will be sufficient. Attribution to the ‘entangled state’ of ‘super-correlations’48 on the basis of the sinusoidal curve is nonsensical; the shape of the curve requires nothing more than Malus’ law operating on pairs in vectorial correlation (discrete in the vOL model, LR bivectors in Clifford algebra treatments70,77-79, or the pairs actualized with aligned vectors in the NL treatment).
The vOL model involves no “hidden variables”. All the information needed to account for the outcome is carried as intrinsic properties of discrete quantized entities; the information arrives with the photon. There is then no need for “conspiracies” to explain the results; the model is immune to
20 closure of communication ‘loop-holes’ (discussed at greater length later). The only requirement is for the behavior at the discriminators to be natural. On the other hand, the full-visibility amplitude de- pends on alignment; the difference in the two approaches as to how that is achieved is a separate is- sue.
6. Why are these conclusions important? The literature is full of claims (cf.12,15,17,19,20,49) that no objective local theory could yield a curve that departs from the limit of ≤2. Indeed, that limit is nowadays the de facto criterion used for evaluation of the success of experiments in supporting the non-local picture, as shown by claims to “…have measured the S parameter… of Bell’s inequalities to be 2 < S < 2.83, thus violating the clas- sical value of 2 by n standard deviations…” or similar41,47,54-56,80. These comparisons, including in recent “loophole-free test of local realism” using electrons60 or photons55,56,59, are more or less worth- less because the target of 2 (or similar55) is irrelevant. Examples in the popular literature that justify the NL picture based on OL models that use scalar dichotomic qualities such as colored socks81, live and dead cats82, hard/soft, black/white, red/green properties28,61,83 etc., or even the polymorphic quan- tum cakes84, serve only to confuse. Such properties are simply inappropriate to the vectorial states involved and could never generate sinusoidal curves. The simulation highlights a general problem with the BCHSH approach, - that Bell’s OL model (and related treatments) are unable to represent a state with the vectorial properties needed in application of Malus’ law. The inequalities so far dis- cussed show only that no model that omits vectors can generate the sinusoidal curves claimed as sup- porting NL expectations. Justification for non-locality must be otherwise demonstrated.
PART B. MAYBE EINSTEIN GOT IT RIGHT A philosophical hurdle. A newcomer engaging with the entanglement community quickly learns that acceptance of the NL case is general and is based on a conviction that Einstein lost the ar- gument with Bohr; quantum uncertainties preclude assignment of intrinsic properties, and therefore require a treatment in which entities evolve in a superposition of indeterminate states, which are actu- alized only on measurement. The strength of the case was apparent in development of the H-atom model; as atomic spectroscopy provided energy levels for electrons, the potentials for occupancy of orbitals were mapped, completed on inclusion of the spin states in the Schrödinger wave equation, and then more widely applied to flesh-out the periodic table. The standard Copenhagen interpreta- tion, and its ancillary orthonormal treatment became deeply embedded in the zeitgeist of quantum
21 physics. Superposition requires a non-local framework, necessarily treated in the wavy domain. Ad- vances over the 85 years since EPR represent an academic heritage through many generations of a success that has spawned Nobel laureates galore, revolutionized physics and chemistry, and fathered many of the innovations that drive our modern economies. Bell’s treatment was brilliantly framed within this tradition, and the experimental validation of his theorem in the '70s and '80s cemented the non-local view, and extended a confidence that similar treatments should also be applied in the wider spatial context. Insofar as the conventional treatment relates to condensed systems and/or atomic scales, I have no argument with the main conclusions from this spectacular record. However, tests based on entangled photons require their evolution to space-like separation before measurement, and non-local effects are then consequential9. Even Bell expressed misgivings about this side of his theorem81, and confidence must now be further eroded by the wonkiness of the stool on losing two of its legs. The ‘failure’ of local realism arises from the fact that its constraints are real, local, and second law com- pliant. Given the OL model he constructed, Bell’s conclusion seemed justified, and his ideas gained traction because that model was uncritically accepted. The zeitgeist trapped him, and apparently the community in general, in a box that excluded Einstein’s model. Despite advances in physics, the same justifications are still applied, so I will first examine the conventional NL case, and then see if any paradox remaining can be resolved in light of progress beyond the Copenhagen interpretation. In the vOL model, quantum scale entities carry properties intrinsic to discrete states, - Ein- stein’s ‘elements of reality’2,83. Einstein’s case is well known from his criticism of the Copenhagen interpretation as incomplete1,85, but this argument had been discounted by von Neumann who proved that “hidden variables” were not needed in the Copenhagen approach. It was from his analysis of von Neumann’s case that Bell became interested in the entanglement debate (see Section 2 below). Bell himself had earlier14 recognized in the context of the “hidden variables” debate that “…if states with prescribed values...could actually be prepared, quantum mechanics would be observably inadequate....”. Such states are now available in the photon pairs generated from PDC sources. The phase-matching is determined (prescribed) by conservation laws, orientation is deter- mined by the pump laser, and, for example in type II PDC, the two emergent beams are empirically demonstrated to be orthogonally polarized. Paradoxically, the manipulation of these populations in state preparation is explicitly based on full knowledge of their determinate properties. In Sagnac in- terferometer applications58,59,86 (see SI, 3 ii) a) - c)), the polarized components of both PDC outputs (from passage clockwise or counter-clockwise through the interferometer) are separated by beam
22 splitters, and used without mixing to provide distinct polarized populations used in measurement. But use of these is forbidden when an indeterminable superposition is invoked for entangled states. Tak- ing to heart Bell’s message that states with “prescribed values” are contrary to QM, in this section I re-examine the conventional NL treatment and its experimental support, and find it wanting.
1. Bell’s third inequality provides a discriminating case My vOL model is simply an extension of Einstein’s idea. In one sense it is trivial, - if a model matches the alignment expected from NL, of course it will give the same outcome. However, its con- sequences have apparently not previously been appreciated. Otherwise, it would have been obvious that the limits of ≤2 could not exclude local realism. Invocation of that limit would then represent de- liberate obfuscation. Since that is anathematic, whatever has been discussed is something different. In Part A, I show that two of the legs claimed as supporting this stool actually provide neither support nor justification for exclusion of local realism. This failure might be expected to worry the entanglement community. Since the limit of ≤2 does not itself exclude local realism, what does? In discussion with colleagues, loss of the two legs has been dismissed as uninteresting, specifically be- cause of Bell’s third inequality. He originally framed this through “…consider the result of a modi- fied theory…in which the pure singlet state is replaced in the course of time by an isotropic mixture of product states…”, for which he suggested a correlation function, (-⅓a∙β). Although somewhat sketchy, this value is a diminished version the vectorial outcome expected under NL predicates (eq. 2), and involves electron pairs evolving in stochastic orientation, leading to a reduced amplitude compared to the full-amplitude curve 13,14,75. In contrast to the model giving the zigzag, the pairs here retain vectorial properties. A similar reduced amplitude is shown for vOL photon pairs in the LR2 curves generated with a stochastic photon source (Fig. 3B). Leggett (personal communication) has suggested a concise expression of the distinction aris- ing from Bell’s third inequality through the following cases, in which θ1 and θ2 are orientations of the fixed and variable polarizers, respectively:
Case 1: For any possible choice of θ1, there exists an OL model, T, such that for all θ2, fT(θ1, θ2) = fNL(θ1, θ2).
Case 2: There exists an OL model, T, such that for any possible choice of θ1, and for all θ2, fT(θ1, θ2) = fNL(θ1, θ2). In framing these cases, Leggett seems to have recognized that the LR0 outcome of my simu- lation demonstrates model T for Case 1; I take this as a partial validation of the conclusions in the first part of the paper. Failure of the vOL treatment to predict model T for Case 2, the full-amplitude
23 rotational invariance, leaves that as the remaining justification for exclusion of local realism. Despite claims from others to the contrary71,77 (see SI, section 3 (iii)), my simulation shows, in agreement with Bell, that constraints from local realism mean that no OL model can predict the NL rotational invariance19,87. Leggett’s distinction omits an important consideration that will figure in further discussion, - the vectorial properties of the photon source apparent from the generating transition. Though ex- cluded, and therefore of no relevance in the NL case, these properties have to be considered in any vectorial realistic model, and any comparison has to include a dissection of their fate in the NL case. For photons carrying intrinsic properties, Case 1 then becomes more highly restricted, limited to the situation in which the photon population is ordered, and its reference frame is aligned with θ1. For Case 2, the outcome predicted remains unconstrained, in the sense that the NL outcome is independ- ent of the initial orientation of frames, or of whether the population is stochastic or ordered. Any pop- ulation of pairs with propensities in dichotomic correlation must give the same full-amplitude rota- tional invariance for any reference frame at the polarizers (see below). However, since in the stochas- tic case, the initial state is clearly disordered, and the outcome observed depends on actualization of an ordered and aligned photon state at the polarizers, to be credible the treatment would have to in- clude a mechanism, including a work-term, to account for the ordering. I argued below that this re- quirement cannot be naturally satisfied.
2. How credible is the NL case for non-locality? In the orthodox NL treatment, the expectation of full-visibility rotational invariance can be framed through a few primary premises: (i) Since Heisenberg uncertainties preclude attribution of intrinsic properties to discrete quan- tum entities, entangled states must be treated as in an indeterminable superposition of all possible states during evolution to the measurement context. (ii) Dichotomic spin states, essential to both the wavefunction and the matrix operations, are assigned propensities such that on operation of the Pauli matrices, the entities are actualized with real vectors in a frame aligned with the discriminator reference frame. (iii) The matrix operations are assumed to represent a physical behavior leading, under experi- mental conditions, to actualization in alignment at the discriminators.
24 Ironically, the first premise sets up the entangled pair in a state from which vectorial infor- mation is excluded. Since correlated vectorial properties, revealed on measurement at space-like sep- arated stations, are essential to the outcome, the processes through which they do become available and aligned, should have raised all those concerns implicit in Einstein’s “incompleteness” argument. Extending the irony, Bell justified his use of the term “hidden variables” in the context of that argu- ment20. However, quantum uncertainties demand a superposition, subsequent application of the or- thonormal treatment provided a consistent resolution, and von Neumann’s formalization88 showed that no “hidden variables” were needed. Although Bell14 in his critique of von Neumann found logi- cal inconsistencies that allowed “hidden variables” in some contexts89, he also found that they did not apply to that QM treatment, and he therefore saw no reason to invoke them in that context. Instead, in a triple-irony, the inherent difficulties were transferred to local realistic theories. The “hidden varia- bles” were needed there only because in setting up his OL model, Bell had stripped the “entangled” state of its vectorial character. The vOL model now introduces a scenario comparing local and non-local perspectives in which both models are vectorial and compliant with seminal QM essentials. In my vOL interpreta- tion, all energy exchanges are necessarily local and quantized. When the vectorial information is car- ried by the photon, properties are intrinsic and explicit, and therefore cannot be “hidden”, and no conspiracies are needed to explain the outcomes simulated. On the other hand, the above premises necessarily leave the conventional NL model still as “incomplete” as it was when Einstein challenged Bohr1,10,85.
3. Indeterminacy and superposition of the entangled state A fundamental tenet of quantum mechanics is that all energy exchanges at that level are quantized. From a research career in photosynthetic mechanisms, it is obvious that at the molecular level interactions of photons with electrons in molecular orbitals involve local exchanges in which energy is conserved, the charge of the electron is inviolate, and photons are neutral. This is in con- trast with the electromagnetic nature of light inherent in Maxwell’s equations, and the interaction of light with harmonic oscillators through the fields of the wave. This distinction is discussed below, but the neutrality of photons makes mechanistic involvement through electromagnetic properties un- tenable, and I will therefore adopt the former perspective as my starting point.
The uncertainty principle and its application
25 Despite the extensive literature, I can see no reason to believe (in the context of entanglement experiments) that quantum uncertainties should exclude treatments in which discrete photons have intrinsic properties. In the conventional treatment, superposition is called for because uncertainties preclude assignment of definite properties to quantum entities. The treatment dates back to the period when the electronic orbital occupancies of the H-atom model were being sorted out, and the minimal ℏ uncertainty derived for electrons by Heisenberg (휎 휎 ≥ in the 3-D case) then precluded assign- ment of definite properties. The argument was applied in a scenario of simultaneous measurement of conjugate variables for position and momentum on a single electron. Can this approach be applied to photons? Although a similar minimal uncertainty has been suggested for photons90, the logic of this approach simply cannot apply. Since detection consumes the photon, it can occur only once; there’s no way to detect a photon twice, so the equivalent experiment could never be attempted. But that is not a problem in the entanglement context because we’re dealing with two different photons, interrogated through refraction, and separately detected at different stations. Measurement involves two distinct components, - interrogation and detection. From the ex- perimentalist’s perspective, although with photons, detection is a one-off event, it does allow one cer- tainty; a recording of the time and place of arrival of the photon. This here-and-now information is all that is directly available from detection, and leaves as a separate issue how to determine other photon properties. To access those, experiments have necessarily examined populations. In principle, the properties of the photons in a population can be determined when the source and pathway of evo- lution are known; a measurement can then be interpreted in terms of information available from the generating transition, and from interrogation during evolution to detection. Interrogation is of interest only when it does not consume the photon. The processes can be selective (transmission through a filter, prism or monochromator), reflective when mirrors are used, or refractive in, for example, use of lenses, HWPs or polarizers. Refractive events are loss-less, so a single photon usually experiences multiple refractive interactions in its path to detection. Useful information can be gleaned from anal- ysis, because, without changing the energy of the selected photons, the path is perturbed in time and/or in space. At a particular frequency (defining the refractive index), refractive behavior probes the remaining property of the photon, the polarization vector. To engage, the vector of the photon must match a vector for electronic displacement, - along the polarizer axis in a polarizer. The proba- bility that a photon will excite a displacement will then be given by Malus’ law. The uncertainty here is in that classical probability term. Since elemental measurements are made on populations, such
26 statistical uncertainties are ironed out in the mean. Epistemologically, the essential point is that be- cause in the entanglement context, two separate photons are detected, and the refractive events are loss-less, quantized interrogation entails only the statistical uncertainty inherent in probabilities of Malus’ law at the elemental level. Bell saw the distinction between the views of Bohr and Einstein as between “…wavy quan- tum states on the one hand, and in Bohr's ‘classical terms’ on the other…” (see Preface81, and 91). From the correspondence principle, either a particulate or a wavy treatment could be used; choice of one must then have an equivalent expression in the other which gives the same result. In the entan- glement context, Bell seemed to equate Einstein’s view with the classical. As a consequence, with NL expectations required a treatment in their wavy guise, with the entangled pair in superposition, and with Bell’s model excluding vectorial properties, consideration of Einstein’s perspective was ex- cluded, even though it was the obvious particulate QM choice. Following Dirac92-94, advances over eight decades in quantum field theory (QFT) have dealt with the particle/wave duality in the wavy realm by coopting Maxwell’s electromagnetic wave treat- ment; different interpretations have led to many different models95 and a massive literature demand- ing a technical expertise beyond my skill-set. However, a simple conclusion requires no such exper- tise. Apart from the claim to have eliminated Einstein’s model, no experimental test has been sug- gested to eliminate any of the others. Since the BCHSH argument did not explicitly invoke Max- well’s approach, further discussion will be deferred to the concluding section. In the simulation, I im- plement the vectorial consequence of energy exchange by momentum transfer, and ambiguities from Maxwellian borrowings are simply bypassed. For entangled photon pairs, the complexities inherent in complete representation in Hilbert space have perhaps also been overhyped. The spatial complexity is restricted because the action vec- tor is orthogonal to the z-axis of propagation. This limits measurement of vectorial parameters to the x, y plane, which simplifies consideration to the unit circle (Fig. 2 and legend). In the conventional NL treatment, the spin states specified in the wave function are used in the matrix operation to gener- ate real vectors for the photons; since the only known vectors are those of the polarizers, actualiza- tion must necessarily depend on reference to the polarizer frame to implement any alignment. The mechanism for alignment itself is discussed extensively below. The photon vectors used in the unit circle projections are aligned, and could be seen as electric or momentum vectors, and either as com- ing from the complex plane, or as the intrinsic vectors of the vOL treatment. Observables returned on
27 taking squares have the same Malus’ law values in either treatment, so the only component of the treatment that discriminates is the implementation of alignment. Practice elsewhere in physics deals in photon populations, but without any prerequisite for such complexities. For example, in determining the history of our universe, modern cosmology in- vokes three simple notions. (i) Measurements on a uniform population can reveal common properties of the discrete photons making it up. (ii) Those properties are intrinsic and determined by the transi- tions in which the photons were generated. The spectra inform us of the local chemistry. The proper- ties may also be modulated by local fields, for example to generate polarization. (iii) Properties are, in principle, conserved on travel over space-like distances. Although frequencies are modified during their journey by relativistic effects, and gravitational refraction can distort the path, these features al- low interrogations that provide, in a measurement context, information about the source and its envi- ronment. Without these assumptions, we could not construct a history of the cosmos. Similar princi- ples underlie all spectroscopic applications, and my simulation of vOL photon populations is based on these same assumptions. If we recognize with EPR that each photon of the pair carries its own vectorial property, the idea of an ontic indeterminacy loses any meaning. I can see no simple model of the photon other than Einstein’s that is compatible with the quantized nature of its interactions, the time and length scales given by its frequency, and with rela- tivistic constraints. Uncertainties are statistical, - an epistemological challenge. A photon can only be detected once, but in the entanglement context we have two of them, and refractive interrogation is loss-less, so the logic of the uncertainties from measurement on a single entity cannot be taken to ne- cessitate a treatment starting with states in superposition.
4. What is known from the generating transition? Under all experimental protocols reported, useful information has been available about the transition generating the initial state. The correspondence principle8 would require that such infor- mation be considered under any QM treatment. Dirac94 suggests that indeterminacy and superposi- tion necessitate a probabilistic treatment, with an outcome that “…expresses itself through the proba- bility of a particular result…being intermediate between the probabilities of the original states…”. However, it is the quantization itself that demands a probabilistic treatment. Probabilities give Malus’ law yields, so the result of a set of pairwise measurements of vectorial correlations should simply generate the statistical outcome expected from such properties, as it does in my simulation. Problems arise not in the vOL case, which, by recognizing discrete properties, allows a natural treatment, but in the NL case, where, contrary to the correspondence principle, vectorial properties are excluded. With
28 a dichotomic Bell-state, all that can then be represented is the pair-wise correlation of spin topologies (the phase difference, or potentialities) implicit in the spin quantum numbers. A mechanism generat- ing specific vectorial properties in the process of actualization must then be invoked to account for the outcome. (a) Cascade sources. With cascade sources, frequencies are known from spectral lines, experi- mental limits for vectorial correlations from conservation of angular momentum are well known; these determinate properties are used in design of protocols. When photons from arc-discharges are used to excite the atomic beam, since orientation of both the atom beam and the photon source are stochastic, orientation of the pairs after excitation must also be stochastic. Their stochastic nature has real consequences. Excitation of the atomic beam by a laser would lead to photoselection36,37. This was demon- strated by polarization detected along the orthogonal y-axis when the laser was polarization along the z axis of propagation36. Photo-selection along the z axis would excite a population of atoms with vec- tors in that direction, but it would be isotropic in the x, y plane of measurement. Experimentally, the singles-counts measured in that plane would then follow the stochastic pattern36, as expected from this analysis. (b) Sources generated by parametric down conversion. Laser excitation of PDC provides well- oriented photon pairs correlated through conservation of energy and angular momentum (phase- matching96); all properties are defined with respect to those of the pump laser54,80,96,97. The photon pairs separate into two populations, orthogonal in orientation, with determinate vectorial properties (H and V are real orientations in the pump reference frame); they are, in Bell’s usage14, prescribed. The behavior of one population is independent of measurement in the other18,80. The PDC output is clearly a state the “with prescribed values” of Bell’s caveat14. Should we not take Bell’s conclu- sion seriously, and worry about the adequacy of the NL approach in that context? PDC sources have been used in all recent photon-based experiments claimed to support the non-local perspective (cf.17,33,41,47,80,97), and knowledge of these determinate properties is exploited in design of experimental protocols. For example, in a seminal paper41 entangled pairs were generated in complementary cones by PDC in a type-II phase-matching BBO crystal excited at 351 nm. Pho- tons at 702 nm were selected from the two intersection points of overlapping H and V cones. Since the Bell state was HV/VH, the photons at either intersection were not pairwise correlated; for any pair, the entangled partner was in the other intersection. After further tweaking by insertion of half-
29 wave plates (‘state-preparation’), the mixed populations were sent to separate stations for polariza- tion analysis, and measurement. In another application54, compensating crystals were used to maximize the output of entan- gled photons from a double-crystal BBO type-1 source, - a pretty exercise in optical design. 80 In a type-II configuration using a different crystal for PDC (periodically poled KTiOPO4) , and pumped at ~400 nm, the co-linear cones at ~800 nm overlapped completely, but the contribu- tions of the two orthogonal orientations could be distinguished by polarizer rotations80. Entangled partners at the chosen frequency were in opposite halves of the overlap, so could be separated using mirrors and irises, allowing a much larger fraction of the population to be tested. A recent variant of this approach86 used the same crystal for PDC, but configured in a Sagnac interferometer. This configuration, discussed in detail in the SI (section 6 Bb, and Fig. SI_1A), was also used in entanglement experiments with the source located in a satellite to test communications loop-holes in58, and in another recent spectacular over-kill in the context of such loopholes59. As dis- cussed in the SI, some features of the Sagnac configuration are far from conventional. In particular, the signal and idler beams were fully polarized when projected to measurement stations, and the H and V photons at each station came from the two separate PDC processes, so neither station sees the mixture in ontic dichotomy implicit in the conventional NL treatment. The general point I want to emphasize is that both in cascade experiments and PDC-based protocols, detailed information is available from knowledge of the source. Manipulation of the “en- tangled” populations in state preparation is clearly based on exploitation of that information; on spe- cific determinate properties of the source, and on classical behavior at refractive elements. Uncer- tainty is introduced from a mixing of two separate populations for which at least some properties are determinate. With well-determined PDC sources, this generates a determinate dichotomic state quite different from the dichotomy by predicate of the orthonormal application. The appearance of indeter- minacy from mixing does not mean that intrinsic properties and their correlations are lost. When in- trinsic, photon properties would be retained, and photons would then behave at polarizers according to Malus’ law80 to generate my vOL outcome. Given that the initial state is determinate, two ontic changes would be required to generate NL expectations; one, after generation of the pair, to a state in superposition in which vectors are lost, and the other, the notoriously vague “reduction of the wave packet”35, to the determinable states, revealed on measurement, with vectors aligned with the polar- izer frame. Absent a mechanism, this is plain silly.
30 5. How is full-amplitude rotational invariance implemented under NL predicates? Two overlapping scenarios need to be examined, the first a mathematical model, and the sec- ond an attempt to relate the model to processes in the real world. The model. Bell used the conventional QM approach13,20 for entangled electron pairs, summa- rized in the matrix operation of eq. 2. In the BCHSH consensus, essentially the same QM approach was applied to photon pairs9,15. As shown by eq. 2, the matrix operations are treated as vectorial, in- volve photons and polarizers at both stations, and an implicit simultaneity of action. As discussed be- low, the spin correlations determine the potentialities (the phase difference), but these are then taken to also indicate propensities. In operation of the Pauli matrices, the propensities in effect determine that the photon pairs become aligned with the reference polarizer. If the photon was H, it will emerge in the ordinary ray, if V in the extraordinary ray, and the partner photon at station 2 will be actualized with the orthogonal vector implicit in the phase difference. The constraints of the matrices are then resolved as observables, - the Malus’ law yields. Shimony provides a useful summary19 of the above; in effect, the alignment of the photon frame with the fixed polarizer implemented in the matrix opera- tion is described “…by substituting the transmission axis of analyzer I for x and the direction perpen- dicular to both z and this transmission axis for y”, where x and y are H and V in the wavefunction equation. To attain the outcome claimed, the primary premises (Section 3) require auxiliary assumptions. 1) Application of Malus’ law requires vectors for photons and polarizers at both stations, so actual- ization would have to occur before the photons reach the polarizers. Since vectorial information for the photons is exclude in the superposition, vectors have to be provided in the process of actu- alization. The fixed polarizer is the only reference frame, and the difference from the variable po- larizer is referred to that zero. The potentialities of the spin states are represented by the binary matrix elements. In effect, the propensities take on a vectorial agency in the matrix operations, and then implement an alignment constrained so as follow Shimony’s prescription above, ac- counting for the outcomes claimed (see Section 6, 3) and 4) below). 2) In recent applications using PDC sources, both polarizers are fixed, and their discriminator func- tion is implemented by using HWPs to rotate the beams before they arrive at the polarizers. This requires a modified treatment to cover the HWP function, - actualization has to occur at or before the discriminating HWPs. However, since similar refractive components are used upstream in state preparation, the question of why actualization does not occur there becomes problematic. 3) The NL outcome depends only on the angle difference between polarizers, σ. Operation of the Pauli matrices implies simultaneous actualization at both stations; in the math this is no problem,
31 because both polarizers are engaged, but in the real world, the two stations are space-like sepa- rated, and information about polarizer settings is only available locally. The question of how sim- ultaneity is achieved then becomes problematic.
6. Relating the model to the real world The math may be elegant, but the physics is not. Several features are missing, mainly because the spatial element is ignored in the math but cannot be avoided in the physics. Firstly, when starting from a stochastic state, or with a misaligned ordered state, an input of work is required to generate the aligned state needed to explain full-visibility. Secondly, if the polarizers are to retain their natural function, photons must have real properties before they arrive there. Actualization in the ordered state would have to precede interaction with the polarizer. If HWPs are used to provide the discrimi- nator function, actualization must be before them. Thirdly, the stations are space-like separated; a value for the vector for the photon at the reference station can be known only at the instant of actual- ization, but that information is needed simultaneously at the other station to allow actualization of the second photon in appropriate correlation. Fourthly, in the mechanism suggested the spin states are assigned propensities with a causal role, but there is no obvious physical basis for this. Analysis of these anomalies reveals a common problem: the premises do not match the properties of the system under study. 1) There is no evidence for any intrinsic vectorial dichotomy in stochastic populations of electrons or photons. The conventional formalism requires an intrinsic dichotomy of spin states. Interpretations involving such states date back to the seminal work in which a beam of sil- ver atoms, each with an unpaired 5s electron, was analyzed using Stern-Gerlach magnets98. The atoms were aligned by the magnets into two well-defined populations; the partition of the atoms was determined by the spins of their electrons. This was interpreted as showing that electrons come in two different spin states, ↑ (up) and ↓ (down). The partitioning into well-defined popula- tions was then interpreted as showing an intrinsic dichotomy representing propensities for parti- tioning. The spin states became aligned with the field of the Stern-Gerlach magnet as determined by their propensities; the up electrons were expected to emerge in the upper population, and the down electrons in the lower population. However, the dichotomy implicit in this assignment can- not be taken as ontic, because, as fundamental particles, all electrons must be the same (cf.99). Expectation of such a dichotomy came from a misunderstanding. The dichotomy of the discrimi- nator function was interpreted as demonstrating an intrinsic dichotomy of the particles. This di-
32 chotomy has become deeply embedded in the standard orthonormal treatment, and the potentiali- ties implicit in the spin quantum numbers have been invested with the vectorial agency of pro- pensities. Although spin may be an intrinsic property of the electron, spin quantum numbers are not. Like all the electronic quantum numbers of the hydrogen atom, they are topological designators. In their atomic context, the topologies of orbital occupancy are defined by the lower quantum numbers, and the electrons in completed orbitals are paired. The spin quantum numbers, s = ±½, determine the relative orientation of the spin states, with potentialities (the phase difference) given by π/2s; the spin axes of a pair then have opposite vectors (↑ and ↓) so that their magnetic effects cancel. This makes sense in the context of electron pairing; they really are entangled through the work term expressed in the additional bond stability, and the orthonormal operations then give the ensemble result implicit in modern quantum chemistry. But such topological conse- quences cannot be in play with unpaired electrons. When unpaired, nuclear and electron spins can be explored experimentally using NMR or EPR (in this context electron paramagnetic resonance, not to be confused with the authors). When placed in a magnetic field, they are then found to distribute into separate populations fol-